Eliminate the arbitrary constant

Question

Can you fully eliminate the arbitary constant to find the differential equation of y=sin(ax+b)

Answer 1

To find the differential equation of a function which involves k arbitrary constants, you will in general have to differentiate k times to get a system of equations from which you can eliminate the constants. In this case, we differentiate to get y' = acos(ax+b) and y'' = −a²sin(ax+b).

Now, an attentive eye will already notice that y'' is just −a² times y. Thus, we have already found our differential equation: y'' + a²y = 0. Because it was a simple case, we didn't have to solve explicitly for 'b' and substitute. In fact, the constant 'a' was not arbitrary here. The second arbitrary constant actually comes from the fact that this is a homogeneous equation, so that a multiple of a solution is another solution. In other words, if we were solving this, 'a' would be given, and the general solution would be:

y(x) = csin(ax+b), where 'b' and 'c' are arbitrary and are determined by the IVP.